Question

In Exercises \(1-8,\) find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=u+a, y=v+a\)

Short answer

The Jacobian \(\partial(x, y) / \partial(u, v)\) is equal to 1.

Step-by-step solution

Compute the partial derivatives

First of all, the partial derivatives of \(x\) and \(y\) with respect to \(u\) and \(v\) need to be calculated. Given the functions, \(x=u+a\) and \(y=v+a\), their partial derivatives are simple to calculate as follows: \[\frac{\partial x}{\partial u} = 1, \frac{\partial x}{\partial v} = 0, \frac{\partial y}{\partial u} = 0, \frac{\partial y}{\partial v} = 1.\]

Construct the Jacobian matrix

Now, with all the partial derivatives calculated, the next step is to construct the Jacobian matrix. A Jacobian matrix should be constructed as:\[\begin{pmatrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{pmatrix}\] Therefore, the Jacobian matrix for the given functions will be: \[\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}\]

Compute the determinant of the Jacobian matrix

Finally, to complete the task, compute the determinant of the Jacobian matrix. The determinant of a 2x2 matrix \[\begin{pmatrix}a & b \\ c & d\end{pmatrix}\] can be calculated as \(ad-bc\). Thus, the determinant of the obtained Jacobian matrix is: \[1*1 - 0*0 = 1.\]